Henri VII of England, first king of the Tudor dynasty, was the grandson of Owen Tudor, a Welsh esquire who married queen Catherine. The latter, when she married Owen Tudor, was Henri V's young widow and Henri VI's mother. Queen Catherine was a French princess of royal blood, daughter to Charles VI and sister to Charles VII. When it came to bragging rights, Henri VII talked about his grandmother and how he descended, thanks to her, from Charlemagne.

Henry VII's father was Edmund, earl of Richmond. The latter was always called Richmond as it was his peerage title. Therefore Henry VII, before becoming king, also bore the name of Richmond and never was he called "Henry Tudor" but by his political enemies who searched to discredit his claim to the throne.

Henry VII's mother, now, was Margaret Beaufort. She descended from John of Gaunt, duke of Lancaster, and Katherine Swynford. Katherine Swynford had been John's lover for many years before she became his wife. However John of Gaunt actually was the founder of the Lancastrian dynasty. Henry IV of England, father to Henry V and grandfather to Henry VI, was John the Gaunt's son from his first marriage to Blanche of Lancaster. We can therefore argue that through his mother, Henry VII was indeed heir to the House of Lancaster, but only if we start to split hair.

C. S. L. Davies published an article called "Tudor: What's in a Name?" (History, 2012, p. 24-42) in which he details the Lancastrian-Tudor filiation and shows why talking about a "Tudor era" is a contemporary invention. As a matter of fact, he even stresses that Henri VIII and Elizabeth were more comfortable with their Yorkist heritage than their obscure Tudor origin. Henri VIII was the embodiement of the Lancastrian-Yorkist reconciliation for his mother, Elizabeth, was daughter to the Yorkist king Edward IV.

Owen Tudor, Owen who? The fact is that historians, novelists and screenwriters love to talk about the Tudor dynasty or the Tudor era as if it used to be a thing or as if it related to some zeitgeist. It wasn't. Though we like to divide history into periods of time that fit the rise and fall of dynasties, people of the sixteenth century didn't share this feeling. Their chronology was divided by the successions of kings. They'd start to count the years everytime a new king rose to the throne. "In the second year of Henry VII's reign..." Etc. Therefore they had no sense to belong to a Tudor era. They were a bit lost when it came to discuss the genealogy of kings. Pretty much like we are now if we didn't have books to fact check everything.

The "House of Tudor" for that matter was not even a concept. Kings and queens had little to no use of their patronyms. They used their peerage titles. Once you were king of England, the rest had no importance. The crown only mattered, not the dynasty. The name of the game was to find every connexion possible with the former kings to legitimate a claim. It means that though we like to talk about the Lancaster, the York and the Tudor, they certainly didn't. They were heir to Edward III and the long gone Plantagenet line, end of story. Davies writes: "We could [...] claim that there was no consolidated view of various changes cohering into a single 'medieval-modern' or 'Plantagenet-Tudor' devide, as is so often assumed."

I would conclude that the Tudors were not the "successors" or the "false successors" of the Lancasters. They obtained the royal title of England and they could not only claim a Lancastrian but also a Yorkist heritage. They were at the very top and nothing else mattered.